MathDB
Problems
Contests
National and Regional Contests
Saudi Arabia Contests
Saudi Arabia BMO TST
2018 Saudi Arabia BMO TST
2018 Saudi Arabia BMO TST
Part of
Saudi Arabia BMO TST
Subcontests
(4)
4
1
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x f (2f (y) - x) + y^2 f (2x - f (y)) =(f (x))^2/x + f (y f (y))
Find all functions
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
such that
x
f
(
2
f
(
y
)
−
x
)
+
y
2
f
(
2
x
−
f
(
y
)
)
=
(
f
(
x
)
)
2
x
+
f
(
y
f
(
y
)
)
x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))
x
f
(
2
f
(
y
)
−
x
)
+
y
2
f
(
2
x
−
f
(
y
))
=
x
(
f
(
x
)
)
2
+
f
(
y
f
(
y
))
, for all
x
,
y
∈
Z
x, y \in Z
x
,
y
∈
Z
,
x
≠
0
x \ne 0
x
=
0
.
3
2
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square-free partition of 2n positive integers into n pairs
The partition of
2
n
2n
2
n
positive integers into
n
n
n
pairs is called square-free if the product of numbers in each pair is not a perfect square.Prove that if for
2
n
2n
2
n
distinct positive integers, there exists one square-free partition, then there exists at least
n
!
n!
n
!
square-free partitions.
\phi (n) is a divisor of n^2+3
Find all positive integers
n
n
n
such that
ϕ
(
n
)
\phi (n)
ϕ
(
n
)
is a divisor of
n
2
+
3
n^2+3
n
2
+
3
.
2
2
Hide problems
f( 2x^3 + f (y)) = y + 2x^2 f (x)
Find all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
2
x
3
+
f
(
y
)
)
=
y
+
2
x
2
f
(
x
)
f( 2x^3 + f (y)) = y + 2x^2 f (x)
f
(
2
x
3
+
f
(
y
))
=
y
+
2
x
2
f
(
x
)
for all real numbers
x
,
y
x, y
x
,
y
.
2018 numbers 1 and -1 around a circle
Suppose that
2018
2018
2018
numbers
1
1
1
and
−
1
-1
−
1
are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all
2018
2018
2018
products is negative. Find all possible values of sum of
2018
2018
2018
given numbers.
1
2
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min n =(2^a - 2^b)/(2^c - 2^d)
Find the smallest positive integer
n
n
n
which can not be expressed as
n
=
2
a
−
2
b
2
c
−
2
d
n =\frac{2^a - 2^b}{2^c - 2^d}
n
=
2
c
−
2
d
2
a
−
2
b
for some positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
radical axis of two circles bisects segment DE
Let
A
B
C
ABC
A
BC
be a triangle with
M
,
N
,
P
M, N, P
M
,
N
,
P
as midpoints of the segments
B
C
,
C
A
,
A
B
BC, CA,AB
BC
,
C
A
,
A
B
respectively. Suppose that
I
I
I
is the intersection of angle bisectors of
∠
B
P
M
,
∠
M
N
P
\angle BPM, \angle MNP
∠
BPM
,
∠
MNP
and
J
J
J
is the intersection of angle bisectors of
∠
C
N
M
,
∠
M
P
N
\angle CN M, \angle MPN
∠
CNM
,
∠
MPN
. Denote
(
ω
1
)
(\omega_1)
(
ω
1
)
as the circle of center
I
I
I
and tangent to
M
P
MP
MP
at
D
D
D
,
(
ω
2
)
(\omega_2)
(
ω
2
)
as the circle of center
J
J
J
and tangent to
M
N
MN
MN
at
E
E
E
. a) Prove that
D
E
DE
D
E
is parallel to
B
C
BC
BC
. b) Prove that the radical axis of two circles
(
ω
1
)
,
(
ω
2
)
(\omega_1), (\omega_2)
(
ω
1
)
,
(
ω
2
)
bisects the segment
D
E
DE
D
E
.